# Properties

 Label 177.2.a.b Level 177 Weight 2 Character orbit 177.a Self dual yes Analytic conductor 1.413 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 177.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.41335211578$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{5} + \beta q^{6} + ( -3 - \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{5} + \beta q^{6} + ( -3 - \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} + ( -2 - \beta ) q^{10} + ( 1 - 2 \beta ) q^{11} + ( 1 - \beta ) q^{12} + ( -5 + 2 \beta ) q^{13} + ( 1 + 4 \beta ) q^{14} + ( 1 - 2 \beta ) q^{15} -3 \beta q^{16} -3 \beta q^{17} -\beta q^{18} -5 \beta q^{19} + ( 3 - \beta ) q^{20} + ( 3 + \beta ) q^{21} + ( 2 + \beta ) q^{22} + ( -4 + \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + ( -2 + 3 \beta ) q^{26} - q^{27} + ( 2 - 3 \beta ) q^{28} + ( 7 + \beta ) q^{29} + ( 2 + \beta ) q^{30} + ( -5 + 9 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( -1 + 2 \beta ) q^{33} + ( 3 + 3 \beta ) q^{34} + ( 1 - 7 \beta ) q^{35} + ( -1 + \beta ) q^{36} + ( -2 - 3 \beta ) q^{37} + ( 5 + 5 \beta ) q^{38} + ( 5 - 2 \beta ) q^{39} + 5 q^{40} + ( 5 - 5 \beta ) q^{41} + ( -1 - 4 \beta ) q^{42} + ( -5 + 6 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( -1 + 2 \beta ) q^{45} + ( -1 + 3 \beta ) q^{46} + ( -9 + 3 \beta ) q^{47} + 3 \beta q^{48} + ( 3 + 7 \beta ) q^{49} + 3 \beta q^{51} + ( 7 - 5 \beta ) q^{52} + ( 5 - 2 \beta ) q^{53} + \beta q^{54} -5 q^{55} + ( 1 - 7 \beta ) q^{56} + 5 \beta q^{57} + ( -1 - 8 \beta ) q^{58} - q^{59} + ( -3 + \beta ) q^{60} + ( -5 - 3 \beta ) q^{61} + ( -9 - 4 \beta ) q^{62} + ( -3 - \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( 9 - 8 \beta ) q^{65} + ( -2 - \beta ) q^{66} + ( 7 - 6 \beta ) q^{67} -3 q^{68} + ( 4 - \beta ) q^{69} + ( 7 + 6 \beta ) q^{70} + ( -3 + 8 \beta ) q^{71} + ( -1 + 2 \beta ) q^{72} + ( -1 - 3 \beta ) q^{73} + ( 3 + 5 \beta ) q^{74} -5 q^{76} + ( -1 + 7 \beta ) q^{77} + ( 2 - 3 \beta ) q^{78} -3 q^{79} + ( -6 - 3 \beta ) q^{80} + q^{81} + 5 q^{82} + ( -1 + \beta ) q^{83} + ( -2 + 3 \beta ) q^{84} + ( -6 - 3 \beta ) q^{85} + ( -6 - \beta ) q^{86} + ( -7 - \beta ) q^{87} -5 q^{88} + ( -7 + 11 \beta ) q^{89} + ( -2 - \beta ) q^{90} + ( 13 - 3 \beta ) q^{91} + ( 5 - 4 \beta ) q^{92} + ( 5 - 9 \beta ) q^{93} + ( -3 + 6 \beta ) q^{94} + ( -10 - 5 \beta ) q^{95} + ( -5 + \beta ) q^{96} + 3 q^{97} + ( -7 - 10 \beta ) q^{98} + ( 1 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} - 5q^{10} + q^{12} - 8q^{13} + 6q^{14} - 3q^{16} - 3q^{17} - q^{18} - 5q^{19} + 5q^{20} + 7q^{21} + 5q^{22} - 7q^{23} - q^{26} - 2q^{27} + q^{28} + 15q^{29} + 5q^{30} - q^{31} + 9q^{32} + 9q^{34} - 5q^{35} - q^{36} - 7q^{37} + 15q^{38} + 8q^{39} + 10q^{40} + 5q^{41} - 6q^{42} - 4q^{43} - 5q^{44} + q^{46} - 15q^{47} + 3q^{48} + 13q^{49} + 3q^{51} + 9q^{52} + 8q^{53} + q^{54} - 10q^{55} - 5q^{56} + 5q^{57} - 10q^{58} - 2q^{59} - 5q^{60} - 13q^{61} - 22q^{62} - 7q^{63} + 4q^{64} + 10q^{65} - 5q^{66} + 8q^{67} - 6q^{68} + 7q^{69} + 20q^{70} + 2q^{71} - 5q^{73} + 11q^{74} - 10q^{76} + 5q^{77} + q^{78} - 6q^{79} - 15q^{80} + 2q^{81} + 10q^{82} - q^{83} - q^{84} - 15q^{85} - 13q^{86} - 15q^{87} - 10q^{88} - 3q^{89} - 5q^{90} + 23q^{91} + 6q^{92} + q^{93} - 25q^{95} - 9q^{96} + 6q^{97} - 24q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−1.61803 −1.00000 0.618034 2.23607 1.61803 −4.61803 2.23607 1.00000 −3.61803
1.2 0.618034 −1.00000 −1.61803 −2.23607 −0.618034 −2.38197 −2.23607 1.00000 −1.38197
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$59$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.a.b 2
3.b odd 2 1 531.2.a.b 2
4.b odd 2 1 2832.2.a.o 2
5.b even 2 1 4425.2.a.t 2
7.b odd 2 1 8673.2.a.k 2
12.b even 2 1 8496.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.b 2 1.a even 1 1 trivial
531.2.a.b 2 3.b odd 2 1
2832.2.a.o 2 4.b odd 2 1
4425.2.a.t 2 5.b even 2 1
8496.2.a.bb 2 12.b even 2 1
8673.2.a.k 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(177))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + 5 T^{2} + 25 T^{4}$$
$7$ $$1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4}$$
$11$ $$1 + 17 T^{2} + 121 T^{4}$$
$13$ $$1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}$$
$17$ $$1 + 3 T + 25 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 13 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 + 7 T + 57 T^{2} + 161 T^{3} + 529 T^{4}$$
$29$ $$1 - 15 T + 113 T^{2} - 435 T^{3} + 841 T^{4}$$
$31$ $$1 + T - 39 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 7 T + 75 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$1 - 5 T + 57 T^{2} - 205 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T + 45 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 15 T + 139 T^{2} + 705 T^{3} + 2209 T^{4}$$
$53$ $$1 - 8 T + 117 T^{2} - 424 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + T )^{2}$$
$61$ $$1 + 13 T + 153 T^{2} + 793 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$1 - 2 T + 63 T^{2} - 142 T^{3} + 5041 T^{4}$$
$73$ $$1 + 5 T + 141 T^{2} + 365 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 3 T + 79 T^{2} )^{2}$$
$83$ $$1 + T + 165 T^{2} + 83 T^{3} + 6889 T^{4}$$
$89$ $$1 + 3 T + 29 T^{2} + 267 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 3 T + 97 T^{2} )^{2}$$